nLab monoidal model category

Idea

A monoidal model category is a model category which is also a closed monoidal category in a compatible way. In particular, its homotopy category inherits a closed monoidal structure, as does the (infinity,1)-category that it presents.

Definition

Definition

A (symmetric) monoidal model category is

• a model category $\mathcal{C}$

equipped with the further structure of

• a closed symmetric monoidal category $(\mathcal{C}, \otimes, I)$

such that the following two compatibility conditions are satisfied:

1. (pushout-product axiom) For every pair of cofibrations $f \colon X \to Y$ and $f' \colon X' \to Y'$, their pushout-product, hence the induced morphism out of the cofibered coproduct over ways of forming the tensor product of these objects

   $$(X \otimes Y') \overset {X \otimes X'} {\amalg} (Y \otimes X') \longrightarrow Y \otimes Y' \,,$$

   is itself a cofibration, which, furthermore, is acyclic (i.e. also a weak equivalence) if $f$ or $f'$ is so.

   (Equivalently this says that the tensor product $\otimes \colon C \times C \to C$ is a left Quillen bifunctor.)

2. (unit axiom) For every cofibrant object $X$ and every cofibrant resolution $\emptyset \hookrightarrow Q I \stackrel{p_I}{\longrightarrow} I$ of the tensor unit $I$, the resulting morphism

   $$Q I \otimes X \stackrel{p_I \otimes X}{\longrightarrow} I\otimes X \stackrel{\simeq}{\longrightarrow} X$$

is a weak equivalence.

Remark

(internal hom Quillen adjunction)

Let $X$ be a cofibrant object, hence $\varnothing \overset{\exists !}{\to} X$ a cofibration.

In this case the pushout-product axiom (Def. ) says that the tensor product functor $X \otimes (-)$ preserves cofibrations and acyclic cofibrations. Since the ambient category is assumed to be closed monoidal category, so that this functor has a right adjoint internal hom $[X,-]$, this means that it is the left Quillen functor in a Quillen adjunction

$$\mathcal{C} \underoverset {\underset{[X,-]}{\longrightarrow}} {\overset{X\otimes(-)}{\longleftarrow}} { \;\;\;\;\; \bot_{\mathrlap{Qu}} \;\;\;\;\; } \mathcal{C}$$

Remark

(meaning of the pushout product axiom)
The pushout-product axiom (above) is stronger than its implied statement in Exp. , while the latter might superficially seem to be all that would reasonably be required of a model-category theoretic version of a monoidal category.

It is at this point that the established tradition to just say “monoidal model category” for the above definition is somewhat misleading, in that Def. is really a model-category theoretic version of the stronger concept of closed monoidal categories (symmetric closed monoidal really, but the symmetry is not the subtle part). The arguably more accurate terminology “closed monoidal model category” for Def. is probably being avoided because Quillen (1967) originally introduced model categories in general under the name “closed model categories” with the adjective “closed” not meant to refer to its use in closed categories…

Namely, the full strength of the pushout-product axiom is needed to imply that not only the tensor product $\otimes$ but also its internal hom $[-,-] \,\colon\, C^{op} \times C \to C$ is homotopically well-behaved, to wit that $[-,-]$ satisfies the pullback-power axiom, which in turn implies that not just $[X,-]$ but also $[-,A]$ are right Quillen functors for all cofibrant objects $X$ and fibrant objects $A$.

In short, the pushout-product axiom is the model-category theoretic way to ensure that not just the tensor product, but also the internal hom as well as their joint relatin in a two-variable adjunction are homotopically meaningful.

Remark

(cofibrant tensor unit implies unit axiom)
As a special case of Rem. : If the tensor unit $I$ happens to be cofibrant, then the unit axiom in def. is already implied by the pushout-product axiom.

(Because then $Q I \to I$ is a weak equivalence between cofibrant objects and such are preserved by functors that preserve acyclic cofibrations, by Ken Brown's lemma.)

Definition

(monoid axiom)

One says that a monoidal model category (Def. ) satisfies the monoid axiom if every morphism that is obtained as a transfinite composition of pushouts of tensor products $X\otimes f$ of acyclic cofibrations $f$ with any object $X$ is a weak equivalence.

(Schwede-Shipley 00, def. 3.3.).

Remark

In particular, the axiom in def. says that for every object $X$ the functor $X \otimes (-)$ sends acyclic cofibrations to weak equivalences.

Properties

Monoidal homotopy category

Abstractly

Proposition

For $(\mathcal{C},\otimes)$ a monoidal model category, def. , then the derived functor $\otimes^L$ of the tensor product makes the homotopy category of the model category itself into a monoidal category, such that the localization functor

$$\gamma \;\colon\; (\mathcal{C},\otimes) \longrightarrow (Ho(\mathcal{C}), \otimes^L)$$

is a lax monoidal functor.

Proof

Let $V$ be a monoidal model category, and consider it as a derivable category in the sense of (Shulman 11, section 8) with $V_Q$ the subcategory of cofibrant objects and $V_R=V$. Then $\otimes :V\times V\to V$ is left derivable, i.e. it preserves the $Q$-subcategories and weak equivalences. Since deriving is pseudofunctorial (and product-preserving) on the 2-category of derivable categories and left derivable functors, it follows immediately that $Ho(V)$ is monoidal; this is (Shulman 11, example 8.13).

Now let $V_0$ denote the category $V$ with its trivial derivable structure: only isomorphisms are weak equivalences, and all objects are both $Q$ and $R$. Then of course $Ho(V_0) = V$, and $V_0$ is also a pseudomonoid in derivable categories. The identity functor $Id : V_0 \to V$ is not left derivable, since it does not preserve $Q$-objects; but it is right derivable, since we took all objects in $V$ to be $R$-objects (ignoring the fibrant objects in the model structure on $V$). Of course $Id$ is strong monoidal, and this monoidality constraint can be expressed as a square in the double category of (Shulman 11) whose vertical arrows are left derivable functors and whose horizontal arrows are right derivable functors; moreover the axioms on a monoidal functor may be expressed using products and double-categorical pasting in this double category. Therefore, it is all preserved by the double pseudofunctor $Ho$; but $Ho(Id) = \gamma : V \to Ho(V)$. The only thing that is not visible to the double category is the invertibility of the monoidal constraint, and hence this is not preserved by the double pseudofunctor; thus $\gamma$ is only lax monoidal.

Explicitly

Proposition

Let $(\mathcal{C}, \otimes, 1)$ be a monoidal model category with cofibrant tensor unit. Then the left derived functor $\otimes^L$ of the tensor product exists

$$\array{ \mathcal{C}_c\times \mathcal{C}_c &\overset{\otimes}{\longrightarrow}& \mathcal{C}_c \\ {}^{\mathllap{\gamma \times \gamma}}\downarrow &\swArrow_{\mu^{-1}}& \downarrow^{\mathrlap{\gamma}} \\ Ho(\mathcal{C})\times Ho(\mathcal{C}) &\overset{\otimes^L}{\longrightarrow}& Ho(\mathcal{C}) }$$

and makes the homotopy category into a monoidal category $(Ho(\mathcal{C}), \otimes^L, \gamma(1))$.

Moreover, the localization functor

$$\gamma \;\colon\; (\mathcal{C}_c, \otimes, 1) \longrightarrow (Ho(\mathcal{C}), \otimes^L , \gamma(1))$$

on the category of cofibrant objects is a strong monoidal functor with structure morphism the inverse of the above natural isomorpmism

$$\mu_{X,Y} \;\colon\; \gamma(X)\otimes^L \gamma(Y) \longrightarrow \gamma(X \otimes Y) \,.$$

Proof

For the left derived functor (def.) of the tensor product

$$\otimes \; \mathcal{C}\times \mathcal{C} \longrightarrow \mathcal{C}$$

to exist, it is sufficient that its restriction to the subcategory

$$(\mathcal{C} \times \mathcal{C})_c \simeq \mathcal{C}_c \times \mathcal{C}_c$$

of cofibrant objects preserves acyclic cofibrations (Ken Brown's lemma, here).

Every morphism $(f,g)$ in the product category $\mathcal{C}_{c}\times \mathcal{C}_{c}$ may be written as a composite of a pairing with an identity morphisms

$$(f,g) \;\colon\; (c_1, d_1) \overset{(id_{c_1},g)}{\longrightarrow} (c_1,d_2) \overset{(f,id_{c_2})}{\longrightarrow} (c_2,d_2) \,.$$

Now since the pushout product (with respect to tensor product) with the initial morphism $(\ast \to c_1)$ is equivalently the tensor product

$$(\ast \to c_1) \Box_{\otimes} g \;\simeq\; id_{c_1} \otimes g$$

and

$$f \Box_{\otimes} (\ast \to c_2) \;\simeq\; f \otimes id_{c_2}$$

the pushout-product axiom (def. ) implies that on the subcategory of cofibrant objects the functor $\otimes$ preserves acyclic cofibrations. (This is why one speaks of a Quillen bifunctor, see also Hovey 99, prop. 4.3.1).

Hence $\otimes^L$ exists.

By the same decomposition and using the universal property of the localization of a category (def.) one finds that for $\mathcal{C}$ and $\mathcal{D}$ any two categories with weak equivalences (def.) then the localization of their product category is the product category of their localizations:

$$(\mathcal{C} \times \mathcal{D})[(W_{\mathcal{C}} \sqcup W_{\mathcal{D}})^{-1}] \simeq (\mathcal{C}[W^{-1}_{\mathcal{C}}]) \times (\mathcal{D}[W^{-1}_{\mathcal{D}}]) \,.$$

With this, the universal property as a localization (def.) of the homotopy category of a model category (thm.) induces associators: Let

$$\array{ \mathcal{C}_c \times \mathcal{C}_c \times \mathcal{C}_c &\overset{((-)\otimes(-))\otimes (-)}{\longrightarrow}& \mathcal{C} \\ {}^{\mathllap{\gamma_{\mathcal{C} \times \mathcal{C} \times \mathcal{C}}}}\downarrow &\swArrow_{\eta}& \downarrow^{\mathrlap{\gamma_{\mathcal{C}}}} \\ Ho(\mathcal{C}) \times Ho(\mathcal{C}) \times Ho(\mathcal{C}) &\overset{((-)\otimes^L(-))\otimes^L (-)}{\longrightarrow}& Ho(\mathcal{C}) } \;\;\;\,,\;\;\;\;\; \array{ \mathcal{C}_c \times \mathcal{C}_c \times \mathcal{C}_c &\overset{ (-) \otimes ( (-) \otimes (-) ) }{\longrightarrow}& \mathcal{C} \\ {}^{\mathllap{\gamma_{\mathcal{C} \times \mathcal{C} \times \mathcal{C}}}}\downarrow &\swArrow_{\eta'}& \downarrow^{\mathrlap{\gamma_{\mathcal{C}}}} \\ Ho(\mathcal{C}) \times Ho(\mathcal{C}) \times Ho(\mathcal{C}) &\overset{ (-) \otimes^L ( (-) \otimes^L (-) ) }{\longrightarrow}& Ho(\mathcal{C}) }$$

be the natural isomorphism exhibiting the derived functors of the two possible tensor products of three objects, as shown at the top. By pasting the second with the associator natural isomorphism of $\mathcal{C}$ we obtain another such factorization for the first, as shown on the left below,

$$(\star) \;\;\;\;\;\; \array{ \mathcal{C}_c \times \mathcal{C}_c \times \mathcal{C}_c &\overset{((-)\otimes(-))\otimes (-)}{\longrightarrow}& \mathcal{C} \\ {}^{\mathllap{=}}\downarrow &\swArrow_{\alpha}& \downarrow^{\mathrlap{=}} \\ \mathcal{C}_c \times \mathcal{C}_c \times \mathcal{C}_c &\overset{ (-) \otimes ( (-) \otimes (-) ) }{\longrightarrow}& \mathcal{C} \\ {}^{\mathllap{\gamma_{\mathcal{C} \times \mathcal{C} \times \mathcal{C}}}}\downarrow &\swArrow_\eta& \downarrow^{\mathrlap{\gamma_{\mathcal{C}}}} \\ Ho(\mathcal{C}) \times Ho(\mathcal{C}) \times Ho(\mathcal{C}) &\overset{ (-) \otimes^L ( (-) \otimes^L (-) ) }{\longrightarrow}& Ho(\mathcal{C}) } \;\;\;\;\;\; \simeq \;\;\;\;\;\; \array{ \mathcal{C}_c \times \mathcal{C}_c \times \mathcal{C}_c &\overset{((-)\otimes(-))\otimes (-)}{\longrightarrow}& \mathcal{C} \\ {}^{\mathllap{\gamma_{\mathcal{C} \times \mathcal{C} \times \mathcal{C}}}}\downarrow &\swArrow_{\eta'}& \downarrow^{\mathrlap{\gamma_{\mathcal{C}}}} \\ Ho(\mathcal{C}) \times Ho(\mathcal{C}) \times Ho(\mathcal{C}) &\overset{((-)\otimes^L(-))\otimes^L (-)}{\longrightarrow}& Ho(\mathcal{C}) \\ {}^{\mathllap{=}}\downarrow &\swArrow_{\alpha^L}& \downarrow^{\mathrlap{=}} \\ Ho(\mathcal{C})\times Ho(\mathcal{C})\times Ho(\mathcal{C}) &\underset{(-)\otimes^L((-)\otimes^L (-))}{\longrightarrow}& Ho(\mathcal{C}) }$$

and hence by the universal property of the factorization through the derived functor, there exists a unique natural isomorphism $\alpha^L$ such as to make this composite of natural isomorphisms equal to the one shown on the right. Hence the pentagon identity satisfied by $\alpha$ implies a pentagon identity for $\alpha^L$, and so $\alpha^L$ is an associator for $\otimes^L$.

The above equation on pasting composites of natural isomorphism is equivalently just the coherence law for a monoidal functor:

$$\array{ (\gamma(X) \otimes^L \gamma(Y)) \otimes^L \gamma(Z) &\overset{\alpha^L_{\gamma(X), \gamma(Y), \gamma(Z)}}{\longrightarrow}& \gamma(X) \otimes^L (\gamma(Y) \otimes^L \gamma(Z)) \\ {}^{\mathllap{\eta'}}\uparrow && \uparrow^{\mathrlap{\eta}} \\ \gamma( (X \otimes Y) \otimes Z ) &\underset{\gamma(\alpha)}{\longrightarrow}& \gamma(X \otimes (Y \otimes Z)) } \,.$$

Examples

Classical homotopy theory

• A nice category of topological spaces with cartesian product and the classical model structure on topological spaces.

• The category of simplicial sets with cartesian product and the classical model structure on simplicial sets.

• The classical model structure on pointed topological spaces or pointed simplicial sets with the smash product of pointed objects.

Simplicial presheaves

If the underlying site has finite products, then both the injective and the projective, the global and the local model structure on simplicial presheaves becomes a cartesian monoidal model category with respect to the standard closed monoidal structure on presheaves.

See at model structure on simplicial presheaves the section Closed monoidal structure.

Monoidal presheaves

More generally, if a small category $\mathcal{S}$ has finite products and $\mathcal{M}$ is a cofibrantly generated symmetric monoidal model category, then the functor category $Func\big(\mathcal{S}^{op}, \mathcal{M}\big)$ with its object-wise monoidal category-structure and with the projective model structure on functors is itself a monoidal model category (Pavlov & Scholbach 2018, inside proof of Prop. 7.9).

Homological algebra and stable homotopy theory

• The category of chain complexes with the usual tensor product of chain complexes carries various monoidal model category structures, see

• the model structure on chain complexes.

With respect to a symmetric monoidal smash product of spectra:

• the model structure on symmetric spectra

• the model structure on orthogonal spectra

• the model structure for excisive functors

(Schwede-Shipley 00, MMSS 00, theorem 12.1 (iii) with prop. 12.3)

The standard example of a monoidal model category whose unit is not cofibrant is the category of EKMM S-modules.

Categorical model structures

• The category Cat with cartesian product and the folk model structure.

• The category Gray of strict 2-categories with the Gray tensor product and the Lack model structure?.

Model structure on $G$-objects

Assumption

Let $\mathcal{E}$ be a category equipped with the structure of

• a closed symmetric monoidal category;

• a monoidal model category;

such that

• the model structure is cofibrantly generated;

• the tensor unit $I$ is cofibrant.

Proposition

Under these conditions there is for each discrete group $G$ the structure of a monoidal model category on the Borel model structure $\mathcal{E}^{\mathbf{B}G}$ of objects in $\mathcal{E}$ equipped with a $G$-action, for which the forgetful functor

$$\mathcal{E}^{\mathbf{B}G} \to \mathcal{E}$$

preserves and reflects fibrations and weak equivalences.

See Berger & Moerdijk (2006) 2.5.

Model structure on monoids

See model structure on monoids in a monoidal model category.

Related concepts

• cartesian closed monoidal model category

• enriched monoidal model category

• monoidal localization

• monoidal Quillen adjunction

• model structure on monoids in a monoidal model category

• model structure on modules in a monoidal model category

References

Textbook accounts:

• Mark Hovey, Chapter 4, Def. 4.2.6, in: Model Categories, Mathematical Surveys and Monographs, 63 AMS (1999) [ISBN:978-0-8218-4361-1, doi:10.1090/surv/063, pdf, Google books]

• Jacob Lurie, Def. A.3.1.2 in: Higher Topos Theory, Annals of Mathematics Studies 170, Princeton University Press 2009 (pup:8957, pdf)

Historically, the first mention of monoidal model categories (without the unit axiom) under this name is in

• William G. Dwyer, Philip S. Hirschhorn, Daniel M. Kan, Model Categories and More General Abstract Homotopy Theory [pdf]

  (Mentioned in passing in Remark 55.10, with no definition given, but the preceding section discusses the pushout product axiom.)

The term “pushout smash poduct” is used for the case of smash products in pointed categories (such as of pointed simplicial sets or symmetric spectra):

• Mark Hovey, Brooke Shipley, Jeff Smith, Symmetric spectra, J. Amer. Math. Soc. 13 (1998) 149-208 [arXiv:math/9801077, doi:10.1090/S0894-0347-99-00320-3]

The general notion of monoidal model categories and their pushout-product axiom appears in:

• Stefan Schwede, Brooke Shipley, Def. 2.1 Algebras and modules in monoidal model categories, Proc. London Math. Soc. 80 2 (2000) 491-511 [arXiv:math/9801082, doi:10.1112/S002461150001220X]

• Mark Hovey, Def. 1.1 in: Monoidal model categories [arXiv:math/9803002v1]

Although the earliest mentions of terminology appear to be these sources indicated above, the notion itself is older. In particular, Hovey (1999), p. 107 credits the definition of a Quillen bifunctor to Dwyer, Hirschhorn, Kan Smith (2004) (which had for years earlier been “in preparation”), and the definition is of course just a small variant of the definition enriched model categories, which in its specialization to simplicial model categories is due to

• Daniel Quillen, chapter II, section 2 of: Homotopical Algebra, Lecture Notes in Mathematics 43, Springer (1967) [doi:10.1007/BFb0097438]

(The corresponding pullback-power axiom is axiom “SM7” in Quillen (1967).)

The unit axiom together with the fact that the homotopy category is monoidal in this case is due to Hovey.

Conditions for the existence of induced monoidal model structure on Reedy model categories:

• Clark Barwick, p. 282 in: On left and right model categories and left and right Bousfield localizations, Homology, Homotopy and Applications 12 2 (2010) 245–320 [doi:10.4310/hha.2010.v12.n2.a9, euclid:1296223884, subsuming: arXiv:0708.2067, arXiv:0708.2832, arXiv:0708.3435]

Some relevant homotopy category background:

• Mike Shulman, Comparing composites of left and right derived functors, New York Journal of Mathematics Volume 17 (2011) 75-125 (arXiv:0706.2868, publisher)

The monoidal structures for a symmetric monoidal smash product of spectra are due to

• Stefan Schwede, Brooke Shipley, Algebras and modules in monoidal model categories Proc. London Math. Soc. (2000) 80(2): 491-511 (pdf)

• Michael Mandell, Peter May, Stefan Schwede, Brooke Shipley, part III of Model categories of diagram spectra, Proceedings London Mathematical Society Volume 82, Issue 2, 2000 (pdf, publisher)

Further variation of the axiomatics is discussed in

• Michael Batanin, Clemens Berger, Homotopy theory for algebras over polynomial monads (arXiv:1305.0086)

The monoidal model structure on the Borel model structure $\mathcal{E}^{\mathbf{B}G}$ is discussed for instance in

• Clemens Berger, Ieke Moerdijk, The Boardman-Vogt resolution of operads in monoidal model categories, Topology 45 (2006) 807-849 [arXiv:math/0502155, doi:10.1016/j.top.2006.05.001]

Relation to symmetric monoidal (infinity,1)-categories (in particular, that every locally presentable symmetric monoidal $(\infty,1)$-category arises from a symmetric monoidal model category) is discussed in

• Thomas Nikolaus, Steffen Sagave, Presentably symmetric monoidal infinity-categories are represented by symmetric monoidal model categories (arXiv:1506.01475)

Monoidal Reedy model structures are discussed in

• Moncef Ghazel, Fethi Kadhi, Reedy diagrams in V-model categories, arXiv:1610.06535, doi:10.1007/s10485-019-09566-w.

On enhancement of monoidal model categories up to strong monoidal Quillen equivalence:

• Dmitri Pavlov, Jakob Scholbach, p. 20 of: Admissibility and rectification of colored symmetric operads, Journal of Topology 11 3 (2018) 559-601 [doi:10.1112/topo.12008, arXiv:1410.5675]

and for monoidal simplicial model categories:

• Haldun Özgür Bayındır, Boris Chorny, Admissible replacements for simplicial monoidal model categories, Algebr. Geom. Topol. 23 (2023) 43-73 [arXiv:2008.00515, doi:10.2140/agt.2023.23.43]